Recently cloud-based graph convolutional network (GCN) has demonstrated great success and potential in many privacy-sensitive applications such as personal healthcare and financial systems. Despite its high inference accuracy and performance on the cloud, maintaining data privacy in GCN inference, which is of paramount importance to these practical applications, remains largely unexplored. In this paper, we take an initial attempt towards this and develop CryptoGCN--a homomorphic encryption (HE) based GCN inference framework. A key to the success of our approach is to reduce the tremendous computational overhead for HE operations, which can be orders of magnitude higher than its counterparts in the plaintext space. To this end, we develop a solution that can effectively take advantage of the sparsity of matrix operations in GCN inference to significantly reduce the encrypted computational overhead. Specifically, we propose a novel Adjacency Matrix-Aware (AMA) data formatting method along with the AMA assisted patterned sparse matrix partitioning, to exploit the complex graph structure and perform efficient matrix-matrix multiplication in HE computation. In this way, the number of HE operations can be significantly reduced. We also develop a co-optimization framework that can explore the trade-offs among the accuracy, security level, and computational overhead by judicious pruning and polynomial approximation of activation modules in GCNs. Based on the NTU-XVIEW skeleton joint dataset, i.e., the largest dataset evaluated homomorphically by far as we are aware of, our experimental results demonstrate that CryptoGCN outperforms state-of-the-art solutions in terms of the latency and number of homomorphic operations, i.e., achieving as much as a 3.10$\times$ speedup on latency and reduces the total Homomorphic Operation Count (HOC) by 77.4\% with a small accuracy loss of 1-1.5$\%$. Our code is publicly available at https://github.com/ranran0523/CryptoGCN.